Theorems on Chord

In this section we will discuss Theorems on Chord.

Theorems on Circles at both the Intermediate and Higher tier are difficult areas for students, which teachers often find equally difficult to deliver.Here, I have explained some Circle theorems based on circle so that student can understand them easily.

1) If two arcs of a circle are congruent, then corresponding chords are equal.Given : PQ = BC Prove that : ∠POQ = ∠BAC

Statements

Reasons

1) BC = PQ

1) Given

2) OP = AB

2) Radius of a circle

3) OQ = AC

3) Radius of a circle

4) ΔPOQ = ΔBAC

4) SSS Postulate

5) ∠POQ = ∠BAC

5) CPCTC

1) The perpendicular from the center of a circle to a chord bisects the chord. Given : PQ is a chord of a circle and OL ⊥ PQ. Prove that : LP = LQ

Statements

Reasons

1) OP = OQ

1) Radii of the same circle.

2) OM = OM

2) Reflexive (common)

3) ∠OMP = ∠OMQ

3) Each 900

4) ΔPMO = ΔQMO

4) HL postulate(RHS)

5) PM = MQ

5) CPCTC

2) If two chords of a circle AB and AC of a circle with center O are such that center O lies on the bisector of ∠BAC, then AB = AC ( chords are equal ). Given : AB = AC Prove that : Center O lies on the bisector of ∠BAC

Statements

Reasons

1) AB = AC

1) Given

2) ∠BAM = ∠CAM

2) Given

3) AM = AM

3) Reflexive (Common)

4) ΔBAM = ΔCAM

4) SAS Postulate

5) BM = CM and ∠BMA = ∠CMA

5) CPCTC

6) ∠BMA + ∠CMA = 900

6) Linear pair angles

7) AM = BM and ∠BMA = ∠CMA = 900

7) From (6)

8) AM is the perpendicular bisector of BC

8) Definition of perpendicular bisector.

9) AM passes through the center O.

9) Perpendicular bisector of a chord always passes through the center.